Optimal. Leaf size=171 \[ \frac{x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac{3 b d^2 e n x^{m+1} (f x)^{m-1}}{4 m^2}-\frac{b d^4 n x^{1-m} \log (x) (f x)^{m-1}}{4 e m}-\frac{b d^3 n x (f x)^{m-1}}{m^2}-\frac{b d e^2 n x^{2 m+1} (f x)^{m-1}}{3 m^2}-\frac{b e^3 n x^{3 m+1} (f x)^{m-1}}{16 m^2} \]
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Rubi [A] time = 0.21477, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2339, 2338, 266, 43} \[ \frac{x^{1-m} (f x)^{m-1} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac{3 b d^2 e n x^{m+1} (f x)^{m-1}}{4 m^2}-\frac{b d^4 n x^{1-m} \log (x) (f x)^{m-1}}{4 e m}-\frac{b d^3 n x (f x)^{m-1}}{m^2}-\frac{b d e^2 n x^{2 m+1} (f x)^{m-1}}{3 m^2}-\frac{b e^3 n x^{3 m+1} (f x)^{m-1}}{16 m^2} \]
Antiderivative was successfully verified.
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Rule 2339
Rule 2338
Rule 266
Rule 43
Rubi steps
\begin{align*} \int (f x)^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx &=\left (x^{1-m} (f x)^{-1+m}\right ) \int x^{-1+m} \left (d+e x^m\right )^3 \left (a+b \log \left (c x^n\right )\right ) \, dx\\ &=\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac{\left (b n x^{1-m} (f x)^{-1+m}\right ) \int \frac{\left (d+e x^m\right )^4}{x} \, dx}{4 e m}\\ &=\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac{\left (b n x^{1-m} (f x)^{-1+m}\right ) \operatorname{Subst}\left (\int \frac{(d+e x)^4}{x} \, dx,x,x^m\right )}{4 e m^2}\\ &=\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}-\frac{\left (b n x^{1-m} (f x)^{-1+m}\right ) \operatorname{Subst}\left (\int \left (4 d^3 e+\frac{d^4}{x}+6 d^2 e^2 x+4 d e^3 x^2+e^4 x^3\right ) \, dx,x,x^m\right )}{4 e m^2}\\ &=-\frac{b d^3 n x (f x)^{-1+m}}{m^2}-\frac{3 b d^2 e n x^{1+m} (f x)^{-1+m}}{4 m^2}-\frac{b d e^2 n x^{1+2 m} (f x)^{-1+m}}{3 m^2}-\frac{b e^3 n x^{1+3 m} (f x)^{-1+m}}{16 m^2}-\frac{b d^4 n x^{1-m} (f x)^{-1+m} \log (x)}{4 e m}+\frac{x^{1-m} (f x)^{-1+m} \left (d+e x^m\right )^4 \left (a+b \log \left (c x^n\right )\right )}{4 e m}\\ \end{align*}
Mathematica [A] time = 0.155689, size = 140, normalized size = 0.82 \[ \frac{(f x)^m \left (12 a m \left (6 d^2 e x^m+4 d^3+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )+12 b m \log \left (c x^n\right ) \left (6 d^2 e x^m+4 d^3+4 d e^2 x^{2 m}+e^3 x^{3 m}\right )-b n \left (36 d^2 e x^m+48 d^3+16 d e^2 x^{2 m}+3 e^3 x^{3 m}\right )\right )}{48 f m^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.242, size = 806, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38314, size = 479, normalized size = 2.8 \begin{align*} \frac{3 \,{\left (4 \, b e^{3} m n \log \left (x\right ) + 4 \, b e^{3} m \log \left (c\right ) + 4 \, a e^{3} m - b e^{3} n\right )} f^{m - 1} x^{4 \, m} + 16 \,{\left (3 \, b d e^{2} m n \log \left (x\right ) + 3 \, b d e^{2} m \log \left (c\right ) + 3 \, a d e^{2} m - b d e^{2} n\right )} f^{m - 1} x^{3 \, m} + 36 \,{\left (2 \, b d^{2} e m n \log \left (x\right ) + 2 \, b d^{2} e m \log \left (c\right ) + 2 \, a d^{2} e m - b d^{2} e n\right )} f^{m - 1} x^{2 \, m} + 48 \,{\left (b d^{3} m n \log \left (x\right ) + b d^{3} m \log \left (c\right ) + a d^{3} m - b d^{3} n\right )} f^{m - 1} x^{m}}{48 \, m^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.47814, size = 452, normalized size = 2.64 \begin{align*} \frac{b d^{3} f^{m} n x^{m} \log \left (x\right )}{f m} + \frac{3 \, b d^{2} f^{m} n x^{2 \, m} e \log \left (x\right )}{2 \, f m} + \frac{b d^{3} f^{m} x^{m} \log \left (c\right )}{f m} + \frac{3 \, b d^{2} f^{m} x^{2 \, m} e \log \left (c\right )}{2 \, f m} + \frac{b d f^{m} n x^{3 \, m} e^{2} \log \left (x\right )}{f m} + \frac{a d^{3} f^{m} x^{m}}{f m} - \frac{b d^{3} f^{m} n x^{m}}{f m^{2}} + \frac{3 \, a d^{2} f^{m} x^{2 \, m} e}{2 \, f m} - \frac{3 \, b d^{2} f^{m} n x^{2 \, m} e}{4 \, f m^{2}} + \frac{b d f^{m} x^{3 \, m} e^{2} \log \left (c\right )}{f m} + \frac{b f^{m} n x^{4 \, m} e^{3} \log \left (x\right )}{4 \, f m} + \frac{a d f^{m} x^{3 \, m} e^{2}}{f m} - \frac{b d f^{m} n x^{3 \, m} e^{2}}{3 \, f m^{2}} + \frac{b f^{m} x^{4 \, m} e^{3} \log \left (c\right )}{4 \, f m} + \frac{a f^{m} x^{4 \, m} e^{3}}{4 \, f m} - \frac{b f^{m} n x^{4 \, m} e^{3}}{16 \, f m^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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